Maybe the choice of preposition depends on the effect you produce on the people in the street - so the Queen would always walk through the street, but an unvisible man would always walk along the street. Am I right?

Maybe the choice of preposition depends on the effect you produce on the people in the street - so the Queen would always walk through the street, but an unvisible man would always walk along the street. Am I right?

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As Alavalen suggests, there's isn't really much difference between along the street and through the street, and gvozd has hit the nail D) on the head in #4.
Along is the more neutral of the two - (вдоль) по улице, and in fact, your Russian метелица could sweep equally well along the street, through the street, up the street or down the street - there's no practical difference in meaning.

Through, on the other hand, is more likely to be used where your progress is slower, it is somehow impeded, slowed down or held up. (But often you could also use along in this sort of context too.)

The Queen walks through the street because there are large crowds of onlookers, she is surrounded by bodyguards, she waves, and stops to talk to people.
An invisible man could walk along the street. If it was a busy street with lots of people and traffic, you might say through the street.

In this story from 'The Guardian', the space shuttle makes its way through the streets of LA (slowly, and lots of onlookers). (Along is also possible here.)
In this story from 'The Independent', the Santas run through the streets of Budapest (lots of onlookers, but along is also possible).
In this story from 'MK News', the woman had her handbag snatched while she was walking along the street (simple movement, there's no suggestion of crowds of people or slow progress, so through wouldn't be used.)

In your example "it was impossible for him to carry the axe through the street in his hands", the implication is that it was unacceptable and his progress would be slow, because people would stop and gape at him, wondering why he's carrying an axe.